A Second-order Cone Relaxation-Based Branch-and-Bound Algorithm for Complex Quadratic Programs on Acyclic Graphs

Abstract

Complex quadratically constrained quadratic programs (QCQPs) with underlying acyclic graph structures have special interests in some important practical applications. In this paper, we propose a new second-order cone relaxation for complex QCQPs, and prove some sufficient conditions under which the proposed relaxation is tight. Then, based on the proposed second-order cone relaxation, a branch-and-bound algorithm is developed. The main feature of the proposed branch-and-bound algorithm is that some complex variables are selected with their bounds on modules or phase angles partitioned in the branching procedure. Numerical results indicate that the proposed branch-and-bound algorithm runs faster than Baron on randomly generated test instances, and is also effective in solving some publicly available test instances of optimal power flow problems.

Appendix A: The convex hull of \({\mathcal {F}}_{ij}\)

In this appendix, we prove the following result: If both $(R_{ij},X_{ij})\in {\mathcal {X}}_{ij}$ and $(X_{ii},X_{jj},R_{ij})\in {\mathcal {B}}_{ij}$ hold, then we have $(X_{ii},X_{jj},R_{ij},X_{ij}) \in \text {Conv}{\mathcal {F}}_{ij}$ and $(X_{ii},X_{jj},X_{ij}) \in \text {Conv}{\mathcal {J}}_{ij}$. This result is an extension of some previous results in [17,18,19]. We provide a proof here for the completeness of the paper.

Based on Corollary 5 in [19], we have ${\mathcal {B}}_{ij}=\text {Conv} {\mathcal {T}}_{ij}$. Besides, for a fixed $R_{ij}\geqslant 0$, the constraint $(R_{ij},X_{ij})\in {\mathcal {X}}_{ij}$ implies that $X_{ij}\in \text {Conv}{\mathcal {Z}}_{ij}(R_{ij})$. Hence, for any $(i,j)\in E$, the two constraints $(X_{ii},X_{jj},R_{ij})\in {\mathcal {B}}_{ij}$ and $(R_{ij},X_{ij})\in {\mathcal {X}}_{ij}$ imply the following decompositions:

$$\begin{aligned} \begin{aligned} (X_{ii},X_{jj},R_{ij})=\sum _{t=1}^r \lambda _t (X^t_{ii},X^t_{jj},R^t_{ij}),~X_{ij}=\sum _{s=1}^k \alpha _s X^s_{ij}, \end{aligned} \end{aligned}$$

(33)

where $(X^t_{ii},X^t_{jj},R^t_{ij})\in {\mathcal {T}}_{ij}$ for $t=1,\cdots ,r$, $X^s_{ij}\in {\mathcal {Z}}_{ij}(R_{ij})$ for $s=1,\cdots ,k$, $\sum _{i=1}^r \lambda _t=1$, $\sum _{s=1}^k \alpha _s=1$, and $\lambda _1,\cdots ,\lambda _r,\alpha _1,\cdots ,\alpha _k\geqslant 0$. Let $\theta ^s_{ij}=\arg (X^s_{ij})$ and denote $X^s_{ij}=R_{ij}\textrm{e}^{ \texttt {i} \theta ^s_{ij}}$ for each $s\in \{1,\cdots ,k\}$. Following (33), together with $\sum _{i=1}^r \lambda _t=1$ and $\sum _{s=1}^k \alpha _s=1$, we have the following results:

$$\begin{aligned} \begin{aligned}&(X_{ii},X_{jj},R_{ij},X_{ij})= \sum _{s=1}^k \sum _{t=1}^r \lambda _t \alpha _s (X^t_{ii},X^t_{jj},R^t_{ij},R^t_{ij} \textrm{e}^{ \texttt {i} \theta ^s_{ij}}),\\&\sum _{s=1}^k \sum _{t=1}^r \lambda _t \alpha _s =\sum _{s=1}^k \left( \sum _{t=1}^r \lambda _t\right) \alpha _s =\sum _{s=1}^k \alpha _s=1. \end{aligned} \end{aligned}$$

(34)

Moreover, since

$$\begin{aligned} (X^t_{ii},X^t_{jj},R^t_{ij},R^t_{ij} \textrm{e}^{ \texttt {i} \theta ^s_{ij}})\in {\mathcal {F}}_{ij},~s=1,\cdots ,k,~t=1,\cdots ,r, \end{aligned}$$

(35)

we have $(X_{ii},X_{jj},R_{ij},X_{ij}) \in \text {Conv}{\mathcal {F}}_{ij}$.

Next, we show that $(X_{ii},X_{jj},X_{ij}) \in \text {Conv}{\mathcal {J}}_{ij}$. Following (34), we have

$$\begin{aligned} \begin{aligned}&(X_{ii},X_{jj},X_{ij})= \sum _{s=1}^k \sum _{t=1}^r \lambda _t \alpha _s (X^t_{ii},X^t_{jj},R^t_{ij} \textrm{e}^{ \texttt {i} \theta ^s_{ij}}),\\&\sum _{s=1}^k \sum _{t=1}^r \lambda _t \alpha _s =1. \end{aligned} \end{aligned}$$

(36)

Note that since $(X^t_{ii},X^t_{jj},R^t_{ij})\in {\mathcal {T}}_{ij}$ implies that $R^t_{ij}=(X^t_{ii}X^t_{jj})^{1/2}$, and $X^s_{ij}\in {\mathcal {Z}}_{ij}(R_{ij})$ implies that $\theta ^s_{ij}=\arg (X^s_{ij})\in {\mathcal {A}}_{ij}$, we have

$$\begin{aligned} (X^t_{ii},X^t_{jj},R^t_{ij} \textrm{e}^{ \texttt {i} \theta ^s_{ij}})\in {\mathcal {J}}_{ij},~s=1,\cdots ,k,~t=1,\cdots ,r. \end{aligned}$$

(37)

Thus, $(X_{ii},X_{jj},X_{ij})$ is in the convex hull of ${\mathcal {J}}_{ij}$.